Beyond Projection: A General Framework for Self-Evolution - From Neural Intelligence to All Domains

V13. 0 - Academic Edition (2026-03-20, Beijing Time / UTC+8) This paper presents a scalable mathematical principle for self-evolution that transcends artificial intelligence and applies to diverse domains requiring autonomous optimization. Through rigorous theoretical analysis and systematic empirical validation across four standard benchmarks (CIFAR-10, CIFAR-100, MedMNIST, TOX21) with varying mechanism counts (K5, K8, K10, K22) and 100 training-inference combinations each, we demonstrate that this framework behaves as predicted by geometric principles, not as a fixed architectural pattern. Core Contribution: Projection-orthogonality duality as a universal principle for self-evolution Four integrated dimensions: Spatial (Projection/Orthogonality), Temporal (Current/Next State), Value (Coherence/Divergence), Embodied (Perception/Action) Mathematical formalization of why dot-product architectures asymptotically approach but never transcend training data boundaries First empirical validation elevating orthogonal generation from domain-specific technique to universal mathematical principle New in V13. 0: Removed CATH 4. 2 results: Protein inverse folding experiment identified data leakage; results are being redesigned with correct protocol. A 1-page core summary version is also provided for easier reading and quick reference New Section: Canonical Application Form of Orthogonal Generation — The general nested serial structure is formalized: op2 (op1 (x, axisA) × layers, axisB) where axisA ⊥ axisB This section clarifies: (1) the correct nested serial form vs. the incorrect parallel mix; (2) domain-specific mappings of orthogonality (image domain: rotation/mirror; algorithm domain: dimensional transpose) ; (3) two recursive extension modes — single-data self-iteration and cross-frame temporal self-evolution; (4) how this general principle applies at any layer of any algorithm — data encoding, hidden states, residual connections, or architectural backbone Predictive Power Demonstrated: Recent independent work (AttnRes, 2026) discovered that replacing fixed residual connections with attention-weighted cross-layer aggregation yields strong empirical gains — operating along the depth axis (L) after the sequence axis (T). This is precisely the nested serial structure on the T⊥L orthogonal axis pair, a direct instantiation of the canonical form described in this framework. The framework not only predicts why such a structure works, but identifies it as one elementary application among a hierarchy of progressively more powerful orthogonal operations — from intra-domain multi-perspective generation, to cross-domain breakthrough, to multi-domain fusion. Engineering discoveries of this kind are the expected outcome when the underlying geometric principle is understood Benchmark Results (V13. 0): CIFAR-10: AUROC 71. 3% → 90. 0% (+18. 7%), FPR@95 86. 2% → 46. 2% (-40. 0%) CIFAR-100: AUROC 68. 3% → 74. 1% (+5. 8%), FPR@95 92. 7% → 82. 9% (-9. 8%) MedMNIST: AUROC 83. 5% → 92. 7% (+9. 2%), Far-OOD reaches 95. 9% TOX21: Near-state-of-the-art results at minimal computational cost Five Key Empirical Findings: Cross-domain universality across natural and medical imaging Task-adaptive scalability (optimal K correlates with task complexity) Real-world applicability (95. 9% Far-OOD detection in medical imaging) Far-OOD advantage as emergent property (inverts conventional expectations) Breaking ID-OOD trade-off (simultaneous improvement in both metrics) Version History: V13. 0 (2026-03-20): Removed CATH 4. 2 (data leakage, being redesigned) ; added canonical application form section formalizing nested serial structure, domain mappings, and recursive extension modes V12. 0 (2026-01-27): Added CATH 4. 2 results (subsequently identified as data leakage) V11. 0 (2026-01-03): Empirical Validation Edition — systematic experiments across three benchmarks, 100 training-inference combinations, Pure ID Training protocol, scalability principle formalization V10. 0 (2026-01-02): Complete Chapter 3 revision, 3-tier orthogonal application hierarchy, formal mathematical tool suite V9. 0 (2025-12-31): Speculative Chapter 10 on Knowledge Field Advection, Xiang correspondence hypothesis V8. 0 (2025-12-30): Yin-Yang complementarity perspective V7. 0 (2025-12-30): Cross-product explanation with Lorentz force analogy V6. 0 (2025-12-29): Academic Edition, complete mathematical formulation with appendices Contact for collaboration: yzb3001313@gmail. com, 382909119@qq. com License: Creative Commons Attribution 4. 0 International (CC BY 4. 0) Note on Originality: This framework represents original theoretical work with first empirical validation. The mathematical duality of projection-orthogonality as a universal self-evolution engine is proposed and now empirically validated here for the first time. Note on Structure: This paper uses author attribution at the conclusion of Chapter 9 to demarcate rigorously-derived and empirically-validated framework (Chapters 1-9) from speculative extension (Chapter 10).